Abstract
In this paper, we establish -analogues of Laplace-type integral transforms by using the concept of -calculus. Moreover, we study some properties of -analogues of Laplace-type integral transforms and apply them to solve some ()-differential equations.
Keywords:
(p,q)-laplace-typed integral transforms; (p,q)-derivative; (p,q)-integral; (p,q)-calculus; (p,q)-difference equations; (p,q)-convolution theorem MSC:
05A30; 44A10; 39A60
1. Introduction
Integral transform techniques are very important for solving many problems in applied mathematics, physics, astronomy, economics and engineering. The integral transform techniques have contributed largely to a variety of theories and applications, such as Laplace, Sumudu, -Integral Laplace, Mohand, Sawi, Kamal and Pourreza transforms. In the sequence of such integral transforms, in 2017, H. Kim [] introduced the Laplace-typed integral transform or -transform, which is defined by
where . The -transform can be applied directly to a suitable problem by choosing appropriately. In Table 1, we list a few of them with their definitions and set for converting -transform into appropriate transforms.

Table 1.
Definitions of some integral transforms.
In 2017, H. Kim [] investigated the solution of Laguerre’s equation by using -transform with , and then, in the same year, H. Kim [] used -transform to solve Volterra integral equation and semi-infinite string. In 2019, S. Sattaso et al. [] studied the properties of Laplace-typed integral transforms for solving differential equations and presented some examples to illustrate the effectiveness of its applicability. In 2020, Y.H. Geum et al. [] applied the matrix expression of convolution and its generalized continuous form with the -transform. Next, in 2021, S.R. Sararha et al. [] introduced the fractional -transform by using modified Riemann–Liouville derivative to solve fractional nonlinear differential equations and applied to the spreading problem of a non-fatal disease within a population.
Quantum calculus, or q-calculus, is referred to as the study of calculus without limits and has also been applied to many areas of mathematics, applied mathematics and physics. It was first studied in the early eighteenth century by a mathematician Euler and developed by Gauss and Ramnaujan. In 1910, F.H. Jackson [,] introduced q-derivative and q-integral, which are known as Jackson derivative and Jackson integral. Many researchers have generalized and developed the q-calculus as found in [,,,,,,,,] and the references cited therein. The book by V. Kac and P. Cheung [] covers the basic theoretical concept of q-calculus.
The topic of q-integral tranform has been scrutinized extensively by many researchers. In 2013, D. Albayrak et al. [] investigated q-analogues of Sumudu transform and derived some properties. In 2014, W.S. Chung et al. [] investigated the q-analogues of the Laplace transform and some properties of the q-Laplace transform. In 2020, S.K.Q. Al-Omari [] proposed the q-analogues and properties of the Laplace-type integral operator in the quantum calculus; see [,,,] for more details.
Post-quantum calculus, or -calculus, is a generalizaion of q-calculus. It was first studied in 1991 by R. Chakrabarti and R. Jagannathan []. In 2013, P.N. Sadjang [] studied the concept of the -derivative, the -integration, -Taylor formulas and the fundamental theorem of -calculus. Many researchers studied and developed the -calculus as found in [,,,,,,,,,,,] and the references cited therein.
Recently, there has been a good deal of extensive research about -integral transforms. In 2017, P.N. Sadjang [] studied the properties of -analogues of the Laplace transform and applied them to solve some -difference equations. In 2019, P.N. Sadjang [] studied the -analogues of the Sumudu transform and gave some properties to solve -difference equations. In 2020, A. Tassaddiq [] proposed -Laplace and -Sumudu transforms with -Aleph-function. The results make a major contribution to the theory of integral transforms and special functions.
Inspired by the above mentioned-literature, we propose to study -analogues of the Laplace-typed integral transform as well as giving some properties that encompass almost all existing -integral transforms and to apply them to solve some ()-differential equations.
The paper is organized as follows: in Section 2, we give some basic knowledge and notation that is used in the next sections; in Section 3, we present some properties of the -analogues of the Laplace-typed integral transform; in Section 4, we apply the -analogues of the Laplace-typed integral transform to some differential equations; and in the last section, we give the conclusion.
2. Preliminaries
In this section, we give basic knowledge that will be used in our work. Throughout this paper, let be constants.
Let us introduce -analogue or -number for , which is defined by
The -factorial is defined by
The -binomial coefficients are defined by
Definition 1
([]). If f is an arbitrary function, then
is the -derivative of the function f.
If in (4), then , which is the q-derivative of the function f; in addition, if in (4), then we get the classical derivative.
Example 1.
Define function by and , where c is a constant; then, we have
Proposition 1.
The -derivatives of the product and quotient rules of functions f and g are as follows:
The proof of Proposition 1 is given in [].
Definition 2.
([]
) If f is an arbitrary function, then the -integral of f on is defined by
If in (7), then (7) reduces to the q-integral of the function f; also, if in (7), then we get the classical integral.
Proposition 2.
If f and g are arbitrary functions, then
is the integration by parts. Note that is allowed.
The proofs of Proposition 2 are given in [].
Definition 3
([]). If , then the -exponential functions are defined by
If in (9) and (10), then we have the q-exponential function []; moreover, if , then (9) and (10) reduce to the classical exponential function.
Proposition 3
([]). If , then the following identities hold:
The proofs of the following Propositions are given in [].
Proposition 4.
If , then
Proposition 5.
If , then the -cosine and the -sine functions are as follows:
Proposition 6.
If , then the -hyperbolic cosine and the -hyperbolic sine functions are as follows:
Proposition 7.
If , then we have
Definition 4
([]). For , the -gamma function is defined by
Definition 5
([]). For , the -beta function is defined by
Theorem 1.
For , the relation between the -gamma function and the -beta function is
The proof of this Theorem is given in [].
3. Properties of -Analogues of Laplace-Typed Integral Transform
In this section, we introduce -analogues of the Laplace-typed integral transform in the form and , which are called -transform of type one and type two, respectively.
Let
and
Now the definition of the -transform of type one and type two is given by:
If and , then (28) and (29) reduce to
and
respectively, which appeared in []. If and , then (28) and (29) reduce to
and
respectively, which appeared in [].
Theorem 2. (Linearity):
If and , then for constants c and d, we have
Proof.
The theorem follows immediately from Definition 6. □
Theorem 3. (Scaling):
If and , then the following formulas hold for non-zero constants β and γ:
Proof.
Using (28) and Proposition 7, we have
Theorem 4.
Let ; then the following formulas hold:
Proof.
Remark 1.
Theorem 5.
If , then the following identities hold:
- (i)
- (ii)
- (iii)
- (iv)
Proof.
We prove by mathematical induction: obviously, is true for . Assuming that is true and using the -integration by parts, we obtain
Remark 2.
If and , then Theorem 5 and reduce to
respectively, which appeared in []. If and , then Theorem 5 and reduce to
respectively, which appeared in [].
Theorem 6.
If , then the following identities hold:
- (i)
- (ii)
- (iii)
- (iv)
Proof.
Remark 3.
If and , then Theorem 6 and reduce to
and
respectively, which appeared in []. Furthermore, if and , then Theorem 6 and reduce to
and
respectively, which appeared in [].
Theorem 7.
If , then the following identities hold:
- (i)
- (ii)
- (iii)
- (iv)
Proof.
Remark 4.
If and , then Theorem 7 reduces to
which appeared in []. Furthermore, if and , then Theorem 7 reduces to
which appeared in [].
Theorem 8.
If , then the following identities hold:
- (i)
- (ii)
- (iii)
- (iv)
Proof.
Remark 5.
If set and , then Theorem 8 reduces to
which appeared in []. Furthermore, if and , then Theorem 8 reduces to
which appeared in [].
Corollary 1.
If , then we have
- (i)
- ;
- (ii)
The proofs of Corollary 1 follow (18) and (20); therefore, the details of Theorems 7 and 8 are omitted.
Theorem 9. (Transforms of integrals):
Let and , then the following identities hold:
- (i)
- (ii)
- (iii)
- (iv)
- (v)
- (vi)
- .
Proof.
We give , and apply the formula of -integration by parts, we obtain
Next, we get
Consequently,
Let ; then we get
After continuing this process, we obtain the sequence
Remark 6.
If , then Theorem 9 reduces to
Furthermore, if , then (36) reduces to the -transform of integrals, which appeared in [].
Theorem 10. (Transforms of derivatives):
If and has the -transform of type one for each , then the transforms of the first, second and n-th derivatives of f can be written in the following forms:
- (i)
- (ii)
- (iii)
Proof.
Applying the equation above with to prove , we get
In , if , it is not difficult to see that
If , we apply the results of by changing to and . We can write
Therefore, the proof is completed. □
The proof of -transform of the type two in (29) is similar to the one for Theorem 10, and is therefore is omitted.
Remark 7.
If and , then Thoerem 10 reduces to
which appeared in [].
Theorem 11. (Derivative of transforms):
For , the following formulas hold:
- (i)
- (ii)
- (iii)
Proof.
Using (28) to prove , we have
Taking -derivative on both sides with respect to , we get
From (37), taking the second -derivative on both sides with respect to to prove , we have
Following the same process, we can prove . Therefore, the proof is completed. □
The proof of -transform of the type two in (29) is similar to the one for Theorem 11 and therefore is omitted.
Remark 8.
If and , then Theorem 11 reduces to
Furthermore, if , then (38) reduces to the derivative of Laplace transform, which appeared in [].
Corollary 2.
If and , then
Proof.
The proof of -transform of the type two in (29) is similar to the one for Corollary 2, but changes to , and therefore is omitted.
Remark 9.
If and , then Theorem 11 and reduce to
and
respectively, which appeared in [].
Theorem 12. (Transforms of the Heaviside function):
For , let
Then, we have
Proof.
Remark 10.
If , then (41) reduces to
Furthermore, if , then (43) reduces to the -transform of Heaviside function, which appeared in [].
Theorem 13. (Transforms of the Dirac delta function):
For , let
If denotes the limit of as , then we have
where δ is the Dirac delta function.
Proof.
If we take the limit of as , then
Remark 11.
If , then (44) reduces to
Furthermore, if , then (46) reduces to the -transform of Dirac delta function, which appeared in [].
Theorem 14. (Convolution theorem):
If and for , then we have
where
Proof.
Using (48), we get
We then change the variables in the equation above by and use (24), which results in the following form:
Hence, we obtain
Therefore, the proof is completed. □
The proof of -transform of the type two in (29) is similar to one for Theorem 14 and therefore is omitted.
Remark 12.
4. Examples
In this section, we solve the -differential equations using the definition and properties of -transform of type one. We consider the -Cauchy problem and two second-order -differential equations.
Example 3.
The -Cauchy problem is in the following form:
where c is a constant.
Applying -transform of the type one, we get
Using the initial conditions , we obtain
Hence
and so
we obtain the solution
In addition, if and , then (49) reduces to
Example 4.
The second order -differential equation is in the following form:
Taking -transform of the type one on both sides, we have
After simplifying the above equation, we get
Hence
and so
The solution is as follows:
Example 5.
Find a solution of
Taking -transform with initial conditions, we have
Hence,
We have the solution:
5. Conclusions
In this work, the properties of the -transform of type one and type two are introduced and proven. After that, we apply the properties of -transform of type one to some -differential equations. The properties proposed and the results of the applications are compared with other papers.
Author Contributions
Conceptualization, S.J., K.N., J.T., S.K.N. and H.K.; Methodology, S.J., K.N., J.T., S.K.N. and H.K.; Formal analysis, S.J., K.N., J.T., S.K.N. and H.K.; writing—original draft preparation, S.J. and K.N.; writing-review and editing, S.J. and K.N.; Visualization, S.J., K.N., J.T., S.K.N. and H.K.; Supervision, J.T., S.K.N. and H.K.; Funding acquisition, K.N. All the authors contributed equally. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Acknowledgments
The first author was supported in part by the Human Resource Development in Science Project (Science Achievement Scholarship of Thailand, SAST).
Conflicts of Interest
The authors declare that they have no competing interests.
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